#947346

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2428	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3020 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)(6,8)(10,11)(14,17)(15,19)(21,22,24,26)(23,25,27,28);(5,6,9,8)(7,10,12,11)(15,19)(18,20)(23,27)(25,28);(5,7,9,12)(6,10,8,11)(13,14,16,17)(15,18,19,20)(21,23)(22,25)(24,27)(26,28);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,22)(23,25)(24,26)(27,28);(5,8,9,6)(7,11,12,10)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297875, #72297898, #72297899
maximal normal subgroup of	#72297875, #72297898, #72297899
maximal quotient of	#72298307

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 854542);