#360541

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#385
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2204 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)(7,12,19,27,20,28,18,31)(8,21,25,30,23,32,24,13)(17,33,35,36,34,29,22,26);(2,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,29)(14,27)(15,31)(16,28)(17,25)(21,26)(23,35)(24,34)(30,33)(32,36);(5,8,9,24,11,23,10,25)(6,13,14,21,16,30,15,32)(7,17,18,22,20,34,19,35)(12,26,27,33,28,36,31,29);(2,4)(8,23)(13,30)(17,34)(21,32)(22,35)(24,25)(26,36)(29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286940, #72286962, #72287084
maximal normal subgroup of	#72286940, #72286962, #72287084

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267737);