#2822326

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3720 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=f,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	#72328304, #72328318, #72328430
maximal normal subgroup of	#72328304, #72328318, #72328430
maximal quotient of	#72330231, #72330590

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2729522);