#2926012

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

3174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fg,d^2=fghi,e^2=fg,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	#72278241
maximal normal subgroup of	#72278241
maximal quotient of	#72278358, #72279004, #72299098

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2833208);