#368507

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250313, #72252980, #72256886
maximal normal subgroup of	#72250313, #72252980, #72256886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275703);