#225995

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133191);