#373634

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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

2504 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)(13,14,18,16,17,20,26,25)(15,21,24,28,23,19,27,22);(2,4)(9,10)(11,12)(13,15)(14,19)(16,22)(17,23)(18,24)(20,21)(25,28)(26,27);(6,8)(10,12)(13,16,18,20,17,25,26,14)(15,22,24,21,23,28,27,19);(2,4)(15,23)(19,21)(22,28)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307889, #72307967, #72308075
maximal normal subgroup of	#72307889, #72307967, #72308075

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280830);