#373862

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#389
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1306 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239822, #72239830, #72239844, …
maximal normal subgroup of	#72239822, #72239830, #72239844, …
maximal quotient of	#72265729

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281058);