#297452

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,13)(8,11)(9,12)(10,15)(14,18)(16,20)(17,19)(21,22,26,30)(23,31,33,35)(24,29,25,28)(27,32,36,34);(1,2,4,3)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,18,19,20)(23,32)(24,25)(27,35)(28,29)(31,36)(33,34);(5,8)(6,11)(7,14,15,16)(9,17)(10,18,13,20)(12,19)(21,23,25,32,26,33,24,34)(22,27,29,35,30,36,28,31);(1,3,4,2)(6,12)(10,13)(11,19)(18,20)(23,33)(27,36)(31,35)(32,34)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ehi,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, 204648);