#392609

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3754 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)(9,10)(11,12)(13,14,18,15,17,20,25,22)(16,19,24,21,23,26,28,27);(2,4)(13,15,18,20,17,22,25,14)(16,21,24,26,23,27,28,19);(6,8)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(10,12)(16,23)(19,26)(21,27)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286545
maximal normal subgroup of	#72286545
maximal quotient of	#72286714, #72286798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299805);