#412518

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi

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, 319714);