#407941

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×#2010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,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	#72265308, #72307302, #72307382
maximal normal subgroup of	#72265308, #72307302, #72307382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315137);