#392227

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3570 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)(15,21)(16,23)(18,25)(19,27)(22,24)(26,28);(6,9)(10,12)(13,14,17,20)(15,18,22,26)(16,19,23,27)(21,25,24,28);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,24)(17,22)(19,28)(20,26)(21,23)(25,27);(16,23)(19,27)(21,24)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299423);