#499267

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

858 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^b=fi,d^c=dh,e^c=ei,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231495, #72252775
maximal normal subgroup of	#72231495, #72252775

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406463);