#414755

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308675, #72308720, #72308786, …
maximal normal subgroup of	#72308675, #72308720, #72308786, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321951);