#624428

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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×#2617	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3427 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)(8,10)(9,12)(13,14)(15,17)(16,19)(18,20)(21,22)(23,27)(24,28)(25,26);(2,4)(5,6,8,10)(7,9,12,11)(14,15)(18,19)(22,26)(23,28);(1,2,3,4)(13,14,17,15)(16,18,20,19)(21,22)(23,24)(25,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,24)(25,28)(26,27);(5,7)(6,9)(8,12)(10,11)(13,16)(14,18)(15,19)(17,20)(21,24)(22,23)(25,27)(26,28)

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326321, #72326324, #72326377
maximal normal subgroup of	#72326321, #72326324, #72326377
maximal quotient of	#72333193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 531624);