#372024

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1173 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)(5,6)(7,11)(8,10)(9,12)(13,14,18,23,19,24,17,20)(15,22,16,21,25,28,26,27);(1,2,4,3)(5,7,8,12)(6,9,10,11)(14,20,24,23)(15,25)(16,26)(21,28,27,22);(7,12)(9,11)(13,15,18,16,19,25,17,26)(14,21,20,22,24,27,23,28);(1,3,4,2)(6,10)(9,11)(13,16,18,25,19,26,17,15)(14,22,20,27,24,28,23,21)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279220);