#397231

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304427);