#363333

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di

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, 270529);