#366312

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
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×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1934 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)(5,6,10,14,11,16,9,15)(7,12,20,28,18,27,19,29)(8,21,23,13,24,31,25,32)(17,33,22,26,34,30,35,36);(2,4)(5,7)(6,12)(8,22,24,35)(9,19)(10,20)(11,18)(13,30,32,33)(14,28)(15,29)(16,27)(17,23,34,25)(21,26,31,36);(5,8)(6,13)(7,17)(9,25)(10,23)(11,24)(12,26)(14,31)(15,21)(16,32)(18,34)(19,35)(20,22)(27,36)(28,30)(29,33);(2,4)(7,18)(12,27)(17,34)(19,20)(22,35)(26,36)(28,29)(30,33)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286324, #72286950, #72287084
maximal normal subgroup of	#72286324, #72286950, #72287084

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273508);