#408255

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,18,20,28)(11,19,22,29,23,33,27,15)(12,24,25,39,26,16,31,32)(21,35,36,30,37,34,40,38);(1,3,4,7)(2,5,6,8)(9,11,20,22,14,23,13,27)(10,15,28,19,18,29,17,33)(12,21,31,36,26,37,25,40)(16,30,39,34,24,38,32,35);(15,29)(19,33)(30,38)(34,35);(2,6)(5,8)(9,12,13,25,14,26,20,31)(10,16,17,32,18,24,28,39)(11,21,27,40,23,37,22,36)(15,30,33,35,29,38,19,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265222, #72307320, #72307382
maximal normal subgroup of	#72265222, #72307320, #72307382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315451);