#2039998

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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

3248 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)(3,4,8,7)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,24)(21,23)(25,26,28,30)(27,29,31,32);(2,6)(4,7)(10,13)(11,15)(17,19,21,24)(18,22,23,20)(27,31)(29,32);(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,29)(28,30)(31,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,29)(28,31)(30,32);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267677, #72267679, #72300833, …
maximal normal subgroup of	#72267677, #72267679, #72300833, …
maximal quotient of	#72267681, #72328353

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1947194);