#372413

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1285 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)(3,4)(5,6)(7,10)(8,9)(11,12)(14,19,22,25)(15,23,24,26);(1,2)(3,4)(13,14,16,25,18,22,17,19)(15,21,26,20,24,27,23,28);(2,4)(6,9)(10,12)(13,15)(14,20)(16,23)(17,26)(18,24)(19,27)(21,25)(22,28);(5,7,8,11)(6,10,9,12)(13,15)(14,21)(16,26)(17,23)(18,24)(19,28)(20,25)(22,27)

Polycyclic presentation:

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

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, 279609);