#7201608

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#36029	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7108804);