#681771

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

3076 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)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18)(19,24)(20,23)(21,22)(25,26,29,30)(27,28,31,32);(2,6)(4,7)(18,22)(23,24)(26,30)(27,31);(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,23,22,24)(25,27)(26,28)(29,31)(30,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,24,22,23)(25,28)(26,27)(29,32)(30,31);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326557, #72326597, #72326635
maximal normal subgroup of	#72326557, #72326597, #72326635
maximal quotient of	#72329061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 588967);