#2032902

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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	C₂×#2847	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3697 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)(6,7)(8,12)(13,14,16,17)(15,18,19,20)(21,22)(23,27)(24,28)(25,26);(6,7)(10,11)(14,17)(15,19)(22,26)(24,27);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,24)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=fgh,c^2=fg,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326390, #72326400, #72326429
maximal normal subgroup of	#72326390, #72326400, #72326429
maximal quotient of	#72328802

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1940098);