#230400

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	#3106	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

520 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137596);