#1118186

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2642 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326549, #72326599, #72326625
maximal normal subgroup of	#72326549, #72326599, #72326625
maximal quotient of	#72328549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1025382);