#7042429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307505, #72307686, #72308503
maximal normal subgroup of	#72307505, #72307686, #72308503

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6949625);