#2199598

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234005, #72234019, #72234031
maximal normal subgroup of	#72234005, #72234019, #72234031

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2106794);