#410861

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(9,10)(11,13)(12,14)(15,16)(17,18,22,30,21,24,20,25)(19,26,29,23,28,32,27,31);(1,2,4,6)(3,7,8,5)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,29)(21,28)(22,27)(24,31)(25,32)(26,30);(1,3,4,8)(2,5,6,7)(18,24)(20,22)(23,31)(27,29);(10,14)(13,16)

Polycyclic presentation:

2^9|a^2=ei,b^2=gh,c^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264431, #72264531, #72264535
maximal normal subgroup of	#72264431, #72264531, #72264535

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318057);