#376002

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1369 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)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,25,28)(16,20,17,21)(23,26,24,27);(6,9)(10,12)(13,15)(14,19)(16,24)(17,23)(18,25)(20,27)(21,26)(22,28);(5,7,8,11)(6,10,9,12)(14,20)(15,23)(16,17)(21,22)(24,25)(26,27);(1,3,4,2)(6,9)(10,12)

Polycyclic presentation:

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

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, 283198);