#375068

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

822 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282264);