#2120991

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2)(3,4)(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,19)(20,24);(1,2,4,3)(9,10,12,14)(11,13,15,16)(17,18,21,19)(20,22,24,23);(2,3)(11,15)(13,16)(17,18,21,19)(20,23,24,22);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22);(1,3,4,2)(9,11,12,15)(10,13,14,16)(17,20,21,24)(18,22,19,23)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=gh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327758, #72327790, #72327887
maximal normal subgroup of	#72327758, #72327790, #72327887

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2028187);