#1269019

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4208 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)(7,12)(8,10)(9,11)(13,14)(15,19)(16,17)(18,20)(21,22,25,27)(23,28,24,26);(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,24)(22,26,27,28);(1,2)(3,4)(21,22)(23,28)(24,26)(25,27);(5,8,9,7)(6,10,11,12)(13,16,17,14)(15,19,20,18)(21,24,25,23)(22,26,27,28);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,28)(25,27)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232636, #72232640, #72232672, …
maximal normal subgroup of	#72232636, #72232640, #72232672, …
maximal quotient of	#72298349, #72328663, #72331028

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1176215);