#6403944

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311140);