#495963

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

616 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)(7,12)(8,10)(9,11)(13,14,18,23,19,20,26,27,17,22,24,28,15,16,21,25);(5,7,9,8)(6,10,11,12)(13,15,17,19)(14,20,22,16)(18,21,24,26)(23,27,28,25);(2,4);(5,8,9,7)(6,10,11,12)(13,16,24,27,19,14,21,28,17,20,18,25,15,22,26,23)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319372, #72319486, #72319524, …
maximal normal subgroup of	#72319372, #72319486, #72319524, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403159);