#304403

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

987 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=efh,d^2=ei,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211599);