#375069

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ei,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, 282265);