#6373184

order	512
nilpotency class	3
isoclinism	isoclinic class of #60367
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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2222 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)(7,12)(8,10)(9,11)(13,14,18,22)(15,24,16,20)(17,23,19,27)(21,25,28,26);(5,7,9,8)(6,10,11,12)(13,15,18,16)(14,20,22,24)(17,26,19,25)(21,23,28,27);(13,16,18,15)(14,21,22,28)(17,26,19,25)(20,23,24,27);(1,3,4,2)(5,8,9,7)(6,10,11,12)(14,22)(20,24)(21,28)(23,27);(13,17,18,19)(14,23,22,27)(15,25,16,26)(20,21,24,28)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,e^b=ei,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280380);