#395774

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1846 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)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,25,22)(16,31,19,30)(20,26)(21,27)(23,29)(24,33,35,32)(34,36);(5,7)(6,11)(8,19)(9,22)(10,18)(12,24)(13,28)(14,25)(15,29)(16,21)(17,23)(20,32)(26,31)(27,35)(30,36)(33,34);(5,8,9,20,10,21,23,34)(6,12,13,26,14,27,29,36)(7,16,17,32,18,19,22,33)(11,24,15,30,25,35,28,31);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322924
maximal normal subgroup of	#72322924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302970);