#855306

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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×#2847	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4149 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326301, #72326316, #72326343, …
maximal normal subgroup of	#72326301, #72326316, #72326343, …
maximal quotient of	#72328492, #72332631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 762502);