#606871

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂×#2678	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3730 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300339, #72300378, #72300412
maximal normal subgroup of	#72300339, #72300378, #72300412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514067);