#814604

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4092 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)(7,11)(8,9)(10,12)(13,14)(15,16)(17,18)(19,20)(21,22,26,25)(23,24,28,27);(2,4)(5,7,9,12)(6,10,8,11)(14,18)(15,20)(21,23,26,28)(22,27,25,24);(5,6,9,8)(7,11,12,10)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,28)(26,27);(2,4)(5,8,9,6)(7,11,12,10)(13,16,17,19)(14,15,18,20)(21,25,26,22)(23,24,28,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=fgi,d^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335395, #72335401, #72335427
maximal normal subgroup of	#72335395, #72335401, #72335427
maximal quotient of	#72336129

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 721800);