#359566

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4066 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261939, #72261949, #72261994
maximal normal subgroup of	#72261939, #72261949, #72261994

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266762);