#367087

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1512 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)(7,8)(9,10,12,14,13,17,20,16)(11,18,21,15,19,22,24,23);(1,2,3,4)(5,6)(7,8)(10,14,17,16)(11,19)(15,18,23,22)(21,24);(2,4)(6,8)(9,10)(11,15)(12,16)(13,17)(14,20)(18,21)(19,23)(22,24);(9,11,20,24,13,19,12,21)(10,15,14,22,17,23,16,18)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216140
maximal normal subgroup of	#72216140
maximal quotient of	#72215962

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274283);