#407978

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285161, #72285600, #72285631
maximal normal subgroup of	#72285161, #72285600, #72285631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315174);