#203742

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=di,e^b=eh,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110938);