#778624

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3400 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:

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

Polycyclic presentation:

2^9|b^2=fgh,c^2=fg,d^2=h,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	#72328877, #72328886, #72328967
maximal normal subgroup of	#72328877, #72328886, #72328967
maximal quotient of	#72330796

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 685820);