#6431798

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431795
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

7300 conjugacy classes of subgroups; only the 986 normal subgroups are shown.

normal subgroups only

This group has 7300 conjugacy classes of subgroups — too many to draw. Only its 986 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6)(7,12)(8,13)(9,14)(10,15)(11,16)(17,24)(18,25)(19,27)(20,26)(21,28)(22,29)(23,30)(31,34)(32,36)(33,35);(5,7,11,10)(6,12,16,15)(8,17,21,20)(9,18,22,23)(13,26,28,24)(14,25,29,30)(19,31,32,33)(27,35,36,34);(2,4)(5,8)(6,13)(7,17)(9,19)(10,20)(11,21)(12,24)(14,27)(15,26)(16,28)(18,31)(22,32)(23,33)(25,34)(29,36)(30,35);(5,9,11,22)(6,14,16,29)(7,18,10,23)(8,19,21,32)(12,25,15,30)(13,27,28,36)(17,31,20,33)(24,34,26,35);(5,10,11,7)(6,15,16,12)(8,20,21,17)(9,18,22,23)(13,26,28,24)(14,25,29,30)(19,31,32,33)(27,34,36,35)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251089, #72251117, #72306379, …
maximal normal subgroup of	#72251089, #72251117, #72306379, …
maximal quotient of	#72310660, C₂×#6431798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338994);