#495452

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

656 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)(7,10)(8,9)(11,12)(13,14,17,21,18,15,22,24,16,20,25,28,26,23,19,27);(2,4)(13,15,17,24,18,20,22,28,16,23,25,27,26,14,19,21);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,20)(15,23)(21,28)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319047, #72319051, #72319066, …
maximal normal subgroup of	#72319047, #72319051, #72319066, …
maximal quotient of	#72322932

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402648);