#353535

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2074 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215768, #72216022, #72216072
maximal normal subgroup of	#72215768, #72216022, #72216072
maximal quotient of	#72215729

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260731);