#6809536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3341	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279530, #72308273
maximal normal subgroup of	#72279530, #72308273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6716732);