#373867

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250027, #72250045, #72250083, …
maximal normal subgroup of	#72250027, #72250045, #72250083, …
maximal quotient of	#72265729, #72343093

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281063);