#2199430

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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3259 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=ghi,c^2=gh,d^2=fgh,e^2=fgh,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	#72232579, #72232586, #72232589, …
maximal normal subgroup of	#72232579, #72232586, #72232589, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2106626);