#2077568

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4014 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)(13,14,18,20)(15,16,17,19)(22,25)(23,27);(5,7,9,8)(6,10,11,12)(13,15,18,17)(14,19,20,16)(22,25)(26,28);(7,8)(10,12)(14,20)(15,17)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,26)(24,25)(27,28);(5,8,9,7)(6,10,11,12)(13,17,18,15)(14,16,20,19)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256722, #72256749, #72256752, …
maximal normal subgroup of	#72256722, #72256749, #72256752, …
maximal quotient of	#72298221

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1984764);