#7182556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 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)(21,22,26,30,25,29,34,28)(23,31,32,36,24,27,33,35);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,27)(24,25)(26,32)(28,36)(29,31)(30,35)(33,34);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(22,28)(23,24)(26,34)(27,35)(29,30)(31,36);(2,4)(7,14)(10,13)(15,20)(18,19)(21,24,25,23)(22,27,29,31)(26,33,34,32)(28,36,30,35);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,24)(27,31)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307373, #72307651, #72307905
maximal normal subgroup of	#72307373, #72307651, #72307905

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089752);