#154271

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61467);