#393939

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3977 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)(11,24)(12,26)(15,28)(16,30)(25,35)(27,33)(29,36)(31,34);(2,4)(5,6)(7,15)(8,12)(9,21)(10,13)(11,18)(14,23)(16,29)(17,28)(19,26)(20,33)(22,24)(25,31)(27,32)(30,36)(34,35);(5,7)(6,11)(8,16)(9,22)(10,17)(12,25)(13,24)(14,28)(15,21)(18,23)(19,30)(20,34)(26,35)(27,36)(29,33)(31,32);(5,8)(6,12)(7,16)(9,20)(10,19)(11,25)(13,26)(14,27)(15,29)(17,30)(18,31)(21,33)(22,34)(23,32)(24,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257114, #72257201, #72257207, …
maximal normal subgroup of	#72257114, #72257201, #72257207, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301135);