#392153

order	512
nilpotency class	3
isoclinism	isoclinic class of #184773
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3862 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299349);