#395783

order	512
nilpotency class	3
isoclinism	isoclinic class of #395781
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1840 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)(5,6,9,14,11,15,10,26)(7,16,19,29,20,28,18,12)(8,21,24,32,25,31,23,13)(17,27,22,36,34,35,33,30);(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,30)(14,28)(15,29)(16,26)(17,24)(21,35)(23,34)(25,33)(27,31)(32,36);(5,8,9,23,11,25,10,24)(6,13,14,31,15,32,26,21)(7,17,19,33,20,34,18,22)(12,27,16,30,29,35,28,36);(2,4)(5,9,11,10)(6,14,15,26)(7,18,20,19)(8,23,25,24)(12,28,29,16)(13,31,32,21)(17,22,34,33)(27,36,35,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286979, #72286985, #72287089
maximal normal subgroup of	#72286979, #72286985, #72287089

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302979);