#373499

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280695);