#367330

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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)(5,6,8,7)(9,10)(11,18)(12,14)(13,17)(15,20)(16,21)(19,22)(23,24);(5,7,8,6)(10,14,17,16)(11,19)(15,18,23,22)(20,24);(1,2,3,4)(9,10)(11,15)(12,16)(13,17)(14,21)(18,20)(19,23)(22,24);(9,11,12,20,13,19,21,24)(10,15,16,18,17,23,14,22)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216137, #72226919, #72226925
maximal normal subgroup of	#72216137, #72226919, #72226925
maximal quotient of	#72216072

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274526);