#415114

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320915, #72321017, #72321073, …
maximal normal subgroup of	#72320915, #72321017, #72321073, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322310);