#378985

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=ef,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi

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, 286181);