#155292

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

5138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2)(3,4)(5,6,7,8)(9,10)(11,12)(13,14,16,15)(17,18,19,20);(1,2,3,4)(9,10,11,12)(17,18,19,20);(5,6)(7,8)(9,10)(11,12)(13,15,16,14)(17,18)(19,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)

Polycyclic presentation:

2^9|a^2=i,b^2=efh,c^2=fi,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	#72278148, #72278184, #72278217, …
maximal normal subgroup of	#72278148, #72278184, #72278217, …
maximal quotient of	C₂×#155292, #72299896, #72299907

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62488);