#370465

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

734 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=efg,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=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, 277661);