#376346

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

981 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)(7,9)(8,10)(11,12)(13,14,19,23)(15,25,21,27)(16,26,17,20)(18,24,28,22);(5,7,8,11)(6,9,10,12)(13,15,19,21)(14,17,23,16)(18,20,28,26)(22,27,24,25);(6,10)(9,12)(13,16)(14,20)(15,22)(17,19)(18,27)(21,24)(23,26)(25,28);(1,3,4,2)(13,17)(14,21)(15,23)(16,19)(18,27)(20,24)(22,26)(25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi,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, 283542);