#373458

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,d^2=fg,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,e^b=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280654);