#2120749

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
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	C₂×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3406 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,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	#72293935, C₂×#2120749
maximal normal subgroup of	#72293935, C₂×#2120749
maximal quotient of	#72293964, C₂×#2120749

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2027945);