#2327073

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fh,d^2=hi,e^2=h,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	#72328131, #72328159, #72328338
maximal normal subgroup of	#72328131, #72328159, #72328338

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2234269);