#6398617

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,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, 6305813);