#395416

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, #72257156, #72257160, …
maximal normal subgroup of	#72257114, #72257156, #72257160, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302612);