#6400632

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307828);