#7186812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1883 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307322, #72307702, #72307914, …
maximal normal subgroup of	#72307322, #72307702, #72307914, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7094008);