#6431797

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

7188 conjugacy classes of subgroups; only the 986 normal subgroups are shown.

normal subgroups only

This group has 7188 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)(7,11)(8,12)(9,13)(10,14)(15,23)(16,24)(17,22)(18,25)(19,26)(20,27)(21,28)(29,35)(30,33)(31,36)(32,34);(11,22)(23,33)(24,34)(35,36);(2,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,23)(13,24)(14,22)(18,29)(19,30)(20,31)(21,32)(25,35)(26,33)(27,36)(28,34);(5,8,10,19)(6,12,14,26)(7,15,17,30)(9,20,21,18)(11,23,22,33)(13,27,28,25)(16,31,32,29)(24,36,34,35);(5,9,10,21)(6,13,14,28)(7,16,17,32)(8,18,19,20)(11,24,22,34)(12,25,26,27)(15,29,30,31)(23,35,33,36)

Polycyclic presentation:

2^9|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	#72251058, #72251117, #72306354, …
maximal normal subgroup of	#72251058, #72251117, #72306354, …
maximal quotient of	#72310626, C₂×#6431797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338993);