#188751

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^b=dhi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95947);