#7171400

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#44881	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,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	#72313813, #72313815, #72313927
maximal normal subgroup of	#72313813, #72313815, #72313927

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7078596);