#1195632

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgh,d^2=hi,e^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1102828);