#1007590

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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	C₂×#2541	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297805, #72297828, #72297839
maximal normal subgroup of	#72297805, #72297828, #72297839
maximal quotient of	#72298309

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 914786);