#300396

order	512
nilpotency class	3
isoclinism	isoclinic class of #300333
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25588	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1003 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fh,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,d^c=dhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207592);