#360262

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2176 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)(9,10,11,12)(13,14,17,20,18,21,16,26)(15,19,24,27,25,28,23,22);(2,4)(5,6)(7,8)(13,15,17,24,18,25,16,23)(14,19,20,27,21,28,26,22);(6,8)(13,14)(15,22)(16,20)(17,26)(18,21)(19,23)(24,28)(25,27);(2,4)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,27,28,22)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308259, #72308269, #72308347
maximal normal subgroup of	#72308259, #72308269, #72308347

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267458);