#2844118

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3305 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)(7,11,12,10)(14,17)(15,19)(21,22)(23,28)(24,26)(25,27);(1,2,3,4)(6,9)(7,12)(13,14,16,17)(15,20,19,18)(21,23)(22,24)(25,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,28)(24,26)(25,27);(1,2,3,4)(5,7)(6,11)(8,12)(9,10)(13,14)(15,20)(16,17)(18,19)(21,22)(23,24)(25,27)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328131, #72328261, #72328314
maximal normal subgroup of	#72328131, #72328261, #72328314
maximal quotient of	#72330114

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2751314);