#789436

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2424	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3468 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)(7,10)(8,11)(9,12)(13,14)(15,19)(16,17)(18,20)(21,22,25,24)(23,27,28,26);(5,7,8,12)(6,9,11,10)(13,15,17,20)(14,18,16,19)(22,24)(26,27);(13,14,17,16)(15,19,20,18)(21,22,25,24)(23,26,28,27);(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,16,17,14)(15,19,20,18)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)(21,24,25,22)(23,26,28,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=fgh,c^2=h,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327766, #72327801, #72327894
maximal normal subgroup of	#72327766, #72327801, #72327894

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 696632);