#364515

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2830 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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.

Relationships

maximal subgroup of	#72234965, #72234970, #72234972
maximal normal subgroup of	#72234965, #72234970, #72234972
maximal quotient of	#72234114

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271711);