#362448

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,18)(16,23)(17,20)(19,22)(21,24)(25,28)(26,27);(5,7,9,8)(6,10,11,12)(14,20,22,15)(21,27,28,23);(5,8,9,7)(6,10,11,12)(13,15,17,22,19,20,18,14)(16,23,24,28,26,27,25,21);(6,11)(10,12)(13,16,18,25,19,26,17,24)(14,21,20,27,22,28,15,23)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216158
maximal normal subgroup of	#72216158
maximal quotient of	#72215768

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269644);