#368080

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 275276);