#184591

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,d^a=dei,c^b=ceh,d^b=dh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91787);