#7173633

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,30)(23,31,34,47)(24,28,37,44)(25,39,32,29)(27,40,42,35)(33,46,36,50)(38,52,48,45)(41,51,43,49);(1,2)(3,4)(7,13)(8,16)(10,17)(11,15);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,32,40,26,34,25,35)(22,27,39,47,30,42,29,31)(24,36,48,49,37,33,38,51)(28,43,52,46,44,41,45,50);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24,26,37)(22,28,30,44)(23,33,34,36)(25,38,32,48)(27,41,42,43)(29,45,39,52)(31,46,47,50)(35,49,40,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,34)(24,37)(27,42)(28,44)(31,47)(35,40)(38,48)(45,52)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7080829);