#2088290

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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3870 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)(5,6,9,8)(7,11,12,10)(13,14,16,17)(15,19,20,18)(22,23)(24,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(22,23)(26,27);(7,12)(10,11)(14,17)(18,19)(21,22,25,23)(24,26,28,27);(1,2)(3,4)(5,8,9,6)(7,10,12,11)(13,15,16,20)(14,18,17,19)(21,23,25,22)(24,27,28,26);(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=f,c^2=fh,d^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	#72327161, #72327236, #72327282, …
maximal normal subgroup of	#72327161, #72327236, #72327282, …
maximal quotient of	#72328988, #72332722

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1995486);