#154710

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2706 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)(5,6,7,8)(13,14,17,15)(16,18,20,19)(22,23)(24,28);(10,12)(14,15)(16,20)(21,22,25,23)(24,27,28,26);(9,10)(11,12)(13,15,17,14)(16,19,20,18)(21,23,25,22)(24,27,28,26);(1,2)(3,4)(5,6)(7,8)(13,16)(14,18)(15,19)(17,20)(21,24,25,28)(22,26,23,27)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72299397, #72299421, #72299492
maximal normal subgroup of	#72299397, #72299421, #72299492
maximal quotient of	#72299914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61906);