#1096638

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

3194 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=gh,e^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	#72330882, #72330910, #72330923
maximal normal subgroup of	#72330882, #72330910, #72330923
maximal quotient of	#72333269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1003834);