#1687401

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2741 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=ghi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232625, #72232691, #72232701
maximal normal subgroup of	#72232625, #72232691, #72232701

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1594597);