#374352

order	512
nilpotency class	3
isoclinism	isoclinic class of #196002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2619	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(2,3)(9,10,13,17,14,16,21,12)(11,15,20,23,19,22,24,18);(1,2,4,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(6,8)(9,12,13,16,14,17,21,10)(11,18,20,22,19,23,24,15);(1,3,4,2)(11,19)(15,22)(18,23)(20,24)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281548);