#368153

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1758 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)(11,14)(12,13)(15,16)(17,18,22,19,21,24,29,26)(20,23,28,25,27,30,32,31);(6,8)(10,13)(14,16)(17,19,22,24,21,26,29,18)(20,25,28,30,27,31,32,23);(2,4)(5,6,7,8)(9,11,12,15)(10,14,13,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32);(10,13)(14,16)(18,24)(19,26)(20,27)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264572, #72264588, #72264590, …
maximal normal subgroup of	#72264572, #72264588, #72264590, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275349);